Some progress has been made on the subject in the past decades.
Under the assumption of zero virtual masses, the Feynman integral of the 1-loop triangle
diagram is formulated as linear combination of the fourth kind of Appell function [1]} whose
arguments are the dimensionless ratios among the external momentum squared, and is simplified
further as the linear combination of the Gauss function \(_2F_1\)  through the quadratic
transformation [2]} in the literature [3]}.
Taking some special assumptions on parameter space, the author of Ref. [4]}
obtains the multiple hypergeometric expressions of the scalar integral \(C_{_0}\)  through the
Mellin-Barnes representations. An algorithm to evaluate the scalar integrals of
the 1-loop vertex-type Feynman diagrams is given in Ref. [5]},
and a geometrical interpretation of the analytic expressions of the scalar integrals
of the 1-loop \(N-\) point diagrams is also presented in Ref. [6]}.
Certainly the expressions of \(C_{_0}\)  function can also be derived from the expressions of
the 1-loop massive \(N-\) point diagrams [7]}, [8]}.
Furthermore Feynman integrals of ladder diagrams with 3 or 4 external lines
can be evaluated through Feynman parametric representation and Mellin-Barnes contour
integrals [9]}.
In fact the 1-loop 2-point function \(B_{_0}\)  can be formulated as the linear combination
of the Gauss function \(_2F_1\)  through the recurrence relations respecting the time-space
dimension, the 1-loop 3-point function \(C_{_0}\)  similarly is presented as the linear
combination of the Appell function \(F_{_1}\) , and the 1-loop 4-point function \(D_{_0}\) 
is given as the linear combination of the Lauricella function
\(F_{_s}\)  [10]}, [11]}, respectively.
The expression of \(C_{_0}\)  is convenient
for analytic continuation and numerical evaluation because continuation of \(F_{_1}\) 
has been analyzed thoroughly. However, how to perform continuation
of the Lauricella function \(F_{_s}\)  outside its convergent domain is still a challenge.
Taking Feynman integrals as generalized hypergeometric functions in dimension regularization,
the authors of Ref. [12]} analyze the Laurent expansion of these hypergeometric
functions around \(D=4\) . The differential-reduction algorithm to evaluate those hypergeometric
functions can be found in Refs. [13]}, [14]}, [15]}, [16]}, [17]}.
A system of linear partial differential equations (PDEs) is given through the corresponding Mellin-Barnes
representation [18]},
and some irreducible master integrals of sunset and bubble Feynman diagrams
are explicitly evaluated through Mellin-Barnes contour [19]}.
In addition, some problems related to the constructions of the \(\varepsilon \) 
expansion of dimensionally regulated Feynman integrals are discussed in
Refs. [20]}, [21]}. Adopting the negative dimensional
integration method, Refs. [22]}, [23]} present the complete massless
and massive 1-loop triangle diagrams results. Proof of the equivalence
between Mellin-Barnes representations and Feynman parametric representations
is given in Ref. [24]}, and the inverse binomial sums relating
to the \(\varepsilon \)  expansion of massive Feynman integrals are presented in
Ref. [25]}. Those results are applied to calculate the
\({\cal O}(\alpha \alpha _{_s})\)  corrections to the relationship between the \(\overline{MS}\) 
mass and the pole of top propagator in the SM [26]}.
